LETTERdoi:10.1038/nature13275

Uplift and seismicity driven by groundwaterdepletion in central CaliforniaColin B. Amos1, Pascal Audet2, William C. Hammond3, Roland Burgmann4,5, Ingrid A. Johanson4 & Geoffrey Blewitt3

Groundwater use in California’s San Joaquin Valley exceeds replen-ishment of the aquifer, leading to substantial diminution of thisresource1–4 and rapid subsidence of the valley floor5. The volumeof groundwater lost over the past century and a half also represents asubstantial reduction in mass and a large-scale unburdening of thelithosphere, with significant but unexplored potential impacts oncrustal deformation and seismicity. Here we use vertical global posi-tioning system measurements to show that a broad zone of rockuplift of up to 1–3 mm per year surrounds the southern San JoaquinValley. The observed uplift matches well with predicted flexure froma simple elastic model of current rates of water-storage loss, most ofwhich is caused by groundwater depletion3. The height of the adja-cent central Coast Ranges and the Sierra Nevada is strongly seasonaland peaks during the dry late summer and autumn, out of phase withuplift of the valley floor during wetter months. Our results suggestthat long-term and late-summer flexural uplift of the Coast Rangesreduce the effective normal stress resolved on the San Andreas Fault.This process brings the fault closer to failure, thereby providing aviable mechanism for observed seasonality in microseismicity atParkfield6 and potentially affecting long-term seismicity rates forfault systems adjacent to the valley. We also infer that the observedcontemporary uplift of the southern Sierra Nevada previously attrib-uted to tectonic or mantle-derived forces7–10 is partly a consequenceof human-caused groundwater depletion.

Hydrospheric mass changes exert direct influence over lithosphericdeformation. Both seasonal and long-term changes to ice, snow or waterloads may induce displacements of the Earth’s surface11–15 and can createstress perturbations that modulate activity on seismogenic faults16–19. Avolume of groundwater approaching approximately 160 km3 in California’sCentral Valley has been lost through pumping, irrigation and evapo-transpiration over the past 150 years or so3,4. Historical and modernrecords demonstrate that groundwater depletion occurs primarily inthe drier, hotter, southern portion of the basin1,3 (the San Joaquin Valley),parallel to the central San Andreas Fault and adjacent to the high topo-graphy of the southern Sierra Nevada (Fig. 1). Previous studies dem-onstrating sensitivity to small-scale stress changes across this sectionof the San Andreas point to seasonal hydrologic6 or temperature20 var-iations to explain observed changes in seismicity rates6 or strain in shal-low boreholes20.

Modest contemporary uplift rates of the southern Sierra Nevada ob-served using space geodesy7–9 have been attributed to various tectonic andgeomorphic drivers, such as buoyant response to mantle delamination10,Basin and Range extensional faulting21, and erosional mass transfer fromPleistocene glaciation22. Inferred stress change from epeirogenic upliftin the southern Sierra Nevada is also potentially linked to the transitionfrom locked to creeping sections of the San Andreas Fault23. Here weseek to explore potential human impacts on contemporary deformationacross this region through global positioning system (GPS) constraintson lithospheric flexure induced by anthropogenic groundwater levelchanges.

Continuous GPS networks spanning the southwestern USA providea high-resolution framework for analysing crustal motion in three dimen-sions (Fig. 1). Longer available records and recently improved processingtechniques enable determination of reliable GPS vertical velocities with aprecision24 of less than a millimetre per year, tied to the Earth system’scentre of mass to within 0.5 mm yr21 (refs 24 and 25). GPS station datawere processed to create individual time series of vertical position andwere then fitted with an empirical model including epoch position, ve-locity, and the amplitude and phase of annual and semiannual har-monic components (to model seasonal effects). This analysis includesonly stations with at least two-and-a-half years of data and vertical rateuncertainties of #1.0 mm yr21 (566 stations). Details of the GPS ana-lysis and processing techniques are included in the Methods.

Figure 1 shows the distribution of vertical GPS rates across centralCalifornia and east of the Sierra Nevada into the Great Basin. Long-termrates exclude GPS stations located within the San Joaquin Valley ground-water basin that display comparatively large signals owing to theirposition on soft sediment. These stations are affected by compaction-driven subsidence through loss of pore water5, poro-elastic deformationrelated to groundwater level fluctuations26, local irrigation and otherprocesses unrelated to solid earth motion.

1Geology Department, Western Washington University, Bellingham, Washington 98225-9080, USA. 2Department of Earth Sciences, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada. 3NevadaGeodetic Laboratory, Nevada Bureau of Mines and Geology and Nevada Seismological Laboratory, University of Nevada, Reno, Nevada 89557, USA. 4Berkeley Seismological Laboratory, University ofCalifornia, Berkeley, California 94720-4760, USA. 5Department of Earth and Planetary Science, University of California, Berkeley, California 97720-4767, USA.

116°118°120°122°

38°

36°

34°

Elevation

(m)

GPS verticaluplift rate(mm yr–1)

4,421

–86

>3

0

<–5

San Joaquin Valley

Ground water level change (m)

~1860–1961

0 to +30 to –12–12 to –24

–24 to –37–37 to –60–60 to –90

–90 to –120

>–120

NV

CA

SJV

SV

Parkfield

Coalinga

SA

F

Sierra N

evada

Sierra N

evada

Figure 2

Figure 2

GreatGreatBasinBasin

Sierra N

evada

Figure 2

GreatBasin

Figure 1 | Contemporary GPS vertical rates and groundwater decline. Mapof vertical uplift rates from GPS stations (circles) spanning California andthe western Great Basin. Stations in the valley showing anomalously largersignals and local irrigation effects are excluded. Contours show historicalchanges in the deep, confined aquifer1. The inset depicts the tectonicconfiguration of the Central Valley groundwater basin (SV, SacramentoValley; SJV, San Joaquin Valley), and the San Andreas Fault (SAF).

0 0 M O N T H 2 0 1 4 | V O L 0 0 0 | N A T U R E | 1

Macmillan Publishers Limited. All rights reserved©2014

Vertical rates across a transect from the central Coast Ranges to therelatively stable Great Basin range up to about 3 mm yr21. The highestaverage velocities concentrate along the margins of the valley (Fig. 2),near the locus of the greatest historical changes in the deep confinedaquifer1 (Fig. 1). The uplift rates of stations west of the San Joaquin Valleyare more variable than those at bedrock sites in the Sierra Nevada,reflecting a higher proportion of stations on or near agricultural basinsor active faults in the central Coast Ranges. Scatter in the vertical ratetransect probably reflects shifts in the locus of groundwater change incomparison with historical averages2, as well as some aleatory variabil-ity. Nearby earthquakes, such as the San Simeon rupture in 2003 ofmoment magnitude 6.5, may also influence stations near the coast withanomalously high rates near 2 mm yr21 (Fig. 2). In the Sierra Nevada,the highest vertical rates occur along the western slope and steadilydecay to the northeast in the Great Basin (Fig. 2). Previous studies showa similarly poor or inverse correlation between elevation and GPS upliftrates across the Sierra Nevada7–9. Modelling of postseismic viscoelasticrelaxation following nearby historical earthquakes suggests that suchtransient strain accounts for only a fraction of the total observed ver-tical signal in the southern Sierra Nevada9.

We compare vertical GPS rates with surface uplift predicted from anelastic half-space model simulating the response to load changes dri-ven by variations in total water storage (Fig. 2) (see Methods). The modelaccounts for surface and subsurface deformation induced by line loadsdistributed across a 60-km-wide strip over the surface of an elastic half-space representing the San Joaquin Valley, the site of long-term, his-torical groundwater unloading. Using a range of elastic parameters, wefitted the GPS-derived vertical velocities with a rate of unloading of8.8 (61.3) 3 107 N m21 yr21 (Fig. 2). We compare this estimate withthe current average unloading rate in the valley based on changes in total

water storage, measured using satellite gravimetry from the GravityRecovery and Climate Experiment (GRACE) (ref. 3; see Methods). Thischange, measured between October 2003 and March 2010, yields an aver-age equivalent unloading rate of 7.2 (62.1) 3 107 N m21 yr21, slightlylower but in good overall agreement with the GPS-derived estimate.

Assuming that changes in total water storage drive vertical motionsurrounding the San Joaquin Valley, the GPS data reflect a combina-tion of short-term, elastic response to ongoing groundwater depletionand longer-term viscous relaxation reflecting the history of hydrosphericmass changes. Slight overestimation of the unloading rate based on theGPS results leaves open the possibility of either a time-dependent (vis-cous) component of the uplift signal, or a (smaller) contribution of on-going tectonic uplift. Both the overall match between the elastic modeland observed GPS uplift (Fig. 2) and seasonal patterns inherent in thephase and amplitude of the GPS vertical time series, however, demon-strate the importance of the instantaneous (elastic) response to ground-water unloading.

Annual peak uplift for GPS stations in the Coast Ranges and SierraNevada occurs in the late summer and early autumn (Fig. 3), correspond-ing with diminished snow and surface water loads27 and overlapping withthe end of the summer growing season and peak groundwater pumpingin the Central Valley (May to September; ref. 2). In contrast, rechargeduring the early spring drives larger-amplitude peak uplift within thevalley through poro-elastic effects of aquifer water levels (Fig. 3 andref. 2). Seasonal vertical displacements are more broadly distributedthan average, long-term trends (Fig. 2), reflecting the variability in totalwater storage in upland catchments feeding the San Joaquin Valley. Nota-bly, stations along the valley margins move upward annually in accord-ance with the Sierra Nevada and Coast Ranges during dry months andshow a corresponding subsidence during the wet winter and spring (Fig. 3).These peripheral stations lead to the bimodal distribution for peak uplifttimes observed for the Central Valley (inset to Fig. 3), indicating thatuplift from hydrospheric load changes dominates even the shortest-term signals in the vertical GPS measurements for stations unaffectedby local irrigation or aquifer effects.

Flexure in the central Coast Ranges due to seasonal hydrospheric un-loading in the San Joaquin Valley provides a viable mechanism to explainthe annual modulation of seismicity on the San Andreas Fault. Both thelocked and creeping fault sections at Parkfield (Fig. 1) exhibit an increasein the number of earthquakes greater than magnitude 1.25 during thelate summer and autumn (Fig. 3), previously attributed to local changesin effective stress linked to the hydrologic cycle6. We explore the poten-tial seasonal impacts of unloading and uplift of the Coast Ranges on short-term changes in the fault-normal stress resolved across the fault using theelastic half-space model (see Methods). Taking estimates of annual varia-tions of 100–300 mm in equivalent water height in the San Joaquin Valleyfrom GRACE data3 and a wider load encompassing the Coast Ranges andthe western Sierra Nevada, the elastic model produces between 3 mm and8 mm of maximum annual vertical ground motion, in good agreementwith the peak-to-peak amplitudes of annual uplift measured by GPS(Fig. 2). The corresponding fault-normal stress variations on the SanAndreas Fault are near 1 kPa at seismogenic depths (Extended DataFig. 4), with peak unclamping during the dry summer and autumn.Although we cannot rule out the potential feedback of reduced effectivenormal stress due to diffusion of pore fluids into fault zone rocks duringrecharge6, our results suggest that unloading may contribute to seasonalmodulation of seismic activity on the central San Andreas Fault. A similarmechanism was invoked to explain annual variations in seismicity in theHimalayas18. Stress changes due to groundwater unloading are somewhathigher for other historically active faults closer to the valley such as theCoalinga thrust system (Fig. 1), where peak-to-peak annual unloadinggives rise to positive Coulomb stress changes of 1.0 kPa to 1.7 kPa (seeMethods and Extended Data Fig. 4).

Given that relatively small annual stress perturbations (around 1 kPa)appear sufficient to modulate earthquakes along the central San AndreasFault6, we use the elastic model to estimate the stress rate caused by

–2

–1

0

1

2

3

Avera

ge G

PS

vert

ical

velo

city (m

m y

r–1)

Ele

vatio

n (km

)

0

1

2

3

4San Joaquin

Valley

Long-term loadSeasonal load200 km

60 km

Max.

Min.

Mean

Elastic model

GPS observation

SAF

CT

0

5

10

15

20

25

30

Annual G

PS

vertic

al

dis

pla

cem

ent (m

m)

0 100 200 300 400 500

Distance (km)

Phase

(Julia

n d

ay)

365

91182273

0Elastic model fit

Mean

Figure 2 | GPS and model comparison. Swath profile of averagecontemporary vertical GPS velocity, annual GPS vertical displacementamplitude, and average topography from the central California Coast Rangeto the western Great Basin (SAF, San Andreas Fault; CT, Coalinga thrust).Vertical velocity and displacement amplitude are shown with 1s uncertainties.The profile includes data from 121 stations and encompasses areas of thegreatest historical and current change to groundwater levels (boxed in Fig. 1).The average GPS velocity is well fitted by an elastic model simulating surfaceuplift resulting from the decline in total water storage (including groundwaterloss) centred along and parallel to the San Joaquin Valley. Seasonal changesin the annual GPS displacement (peak-to-peak amplitude) are distributedmore broadly over the San Joaquin drainage basin, reflecting distribution ofwinter precipitation, snow load and reservoirs. Blue bars show the width ofboth the long-term and the seasonal loads used in the elastic model.

RESEARCH LETTER

2 | N A T U R E | V O L 0 0 0 | 0 0 M O N T H 2 0 1 4

Macmillan Publishers Limited. All rights reserved©2014

ongoing human-induced groundwater removal (see Methods). Takingan estimate of the unloading rate due to loss in water storage in the SanJoaquin Valley of 8.8 (61.3) 3 107 N m21 yr21, we calculate a rate ofunclamping of 0.07–0.18 kPa yr21 for the San Andreas Fault. Coulombfailure stress rates on the Coalinga thrusts are higher, averaging 0.13–0.57 kPa yr21. Estimates of total groundwater loss from historical andmore recent water level records (around 160 km3; refs 3 and 4) corre-spond to a reduction in the effective normal stress of 2.7–9.5 kPa for theSan Andreas Fault, and a Coulomb stress change of 10–15 kPa at Coalinga,since the beginning of groundwater extraction in 1860. These resultssuggest that human activity may give rise to a gradual increase in therate of earthquake occurrence, as suggested by earthquake cataloguesin central California6.

Future scenarios for groundwater in California suggest increasingdemand for agricultural, urban and environmental use2. Climate changewill probably exacerbate the stress on this resource through alteredprecipitation patterns, more frequent droughts, earlier snowmelt, lar-ger floods, and increasing temperatures and evapotranspiration28,29. Wedemonstrate how long-term and seasonal hydrospheric mass changesdue to groundwater removal significantly affect regional crustal defor-mation, which can lead to annual variations in small-earthquake fre-quency and longer-term changes of stresses on the San Andreas Fault.Our model explains recent vertical uplift rates previously attributed toepeirogenic deformation in central California10, indicating that muchof this signal results from anthropogenic groundwater removal. Takentogether, our results highlight new and underappreciated links betweenhuman activity and solid earth processes driven by hydrospheric change.

METHODS SUMMARYElastic model. A two-dimensional model accounts for deformation from a normalline load distributed over a finite-width strip on the surface of the half-space13,30.The vertical displacement rate _u at the surface (z 5 0) along a profile perpendicularto the load centred at x 5 xd is given by:

_u x,z~0ð Þ~ 1{nð Þ _N0

pG 2að Þ 2az x{xd{að Þ ln x{xd{aj j½

{ x{xdzað Þ ln x{xdzaj j�zK

where x and z are the horizontal and vertical dimensions, respectively, _N0 is the rateof line load, n is Poisson’s ratio, G is the shear modulus, a is the strip half-width andK is a far-field rate offset correction. The load is centred on the middle of the valleyand covers its entire width (2a 5 60 km). Given a range of elastic parameters (n5

0.25, G 5 35 6 5 GPa), we fit observed vertical velocities using a line-load rate with_N0 5 8.8 (61.3) 3 107 N m21 yr21 (Fig. 2).

Unloading from groundwater loss. The Sacramento and San Joaquin basins lostan estimated volume of 30.9 (62.6) km3 in total water storage between October2003 and March 2010 (ref. 3). Total water storage change in the San Joaquin Valley(21.5 6 6.2 km3) equates to an unloading of 3.3 (61.0) 3 1013 N yr21 over a valleylength of 450 km, giving an equivalent line load of _N0 5 7.2 (62.1) 3 107 N m21 yr21,in agreement with the elastic model.Unclamping of the San Andreas Fault. The unclamping rate (change in faultnormal stress) on a vertical fault is calculated as30:

_txx~_N0

2pah1{h2ð Þz sin h1{h2ð Þ cos h1zh2ð Þ½ �

where h1 and h2 are the angles from the load edges to any point at depth, measuredclockwise from the positive x direction. For the San Andreas Fault, located about70 km from the 60-km-wide load centre, we estimate an unclamping rate of 0.07–0.18 kPa yr21 at seismogenic depths (5–15 km) from groundwater unloading.

Online Content Any additional Methods, Extended Data display items and SourceData are available in the online version of the paper; references unique to thesesections appear only in the online paper.

Received 24 October 2013; accepted 30 March 2014.

Published online 14 May 2014.

1. Williamson,A.K., Prudic,D.E.&Swain, L. A.Ground-water flow in theCentral Valley,California. US Geol. Surv. Prof. Pap. 1401-D (1989).

2. Faunt, C. C. (ed.) Groundwater availability of the Central Valley aquifer, California.US Geol. Surv. Prof. Pap. 1766 (2009).

3. Famiglietti, J. S. et al. Satellites measure recent rates of groundwater depletion inCalifornia’s Central Valley. Geophys. Res. Lett. 38, L03403 (2011).

4. Scanlon, B. R. et al. Groundwater depletion and sustainability of irrigation inthe US High Plains and Central Valley. Proc. Natl Acad. Sci. USA 109, 9320–9325(2012).

5. Poland, J. F., Lofgren, B. E., Ireland, R. L. & Pugh, R. G. Land subsidence in the SanJoaquin Valley, California, as of 1972. US Geol. Surv. Prof. Pap. 437-H (1975).

6. Christiansen, L. B., Hurwitz, S. & Ingebritsen, S. E. Annual modulation of seismicityalong the San Andreas Fault near Parkfield, CA. Geophys. Res. Lett. 34, L04306(2007).

Phase (Julian day)

Amplitude(mm)

365

91

182

273

0

< 2.0

6.0–11.9

4.0–5.9

2.0–3.9

> 12.0

Peak vertical position

Coast Range

Central Valley

Sierra Nevada

Winter/ spring

Summer/ autumn

Relative regional GPSresponse by seasonup

down

Seismic rate peaks at Parkfield

(Locked, Creeping)

Coast Range

Sierra NevadaCentral Valley

Total

Month

Num

ber

of

sta

tio

ns

1 2 3 4 5 6 7 8 9 10 11 12

10

0

20

30

40

50

60

70

80

90Peak GPS uplift by month

Figure 3 | Seasonal peak uplift from GPS. Phase and peak-to-peak amplitudeof annual vertical GPS displacement for all stations included in our analysis.The inset shows histograms of peak uplift phase (binned by month) forstations in the San Joaquin Valley, Sierra Nevada, and Coast Range. Peaks inseismicity rate for the locked and creeping San Andreas Fault at Parkfieldare defined as months with higher than average declustered seismicity6.Smaller-amplitude peak uplift in the Sierra Nevada and Coast Range during

the dry summer and autumn is largely out of phase with larger-amplitudeuplift in the San Joaquin Valley driven by the poro-elastic response to rechargeduring the wetter winter and spring. Stations peripheral to the valley showuplift patterns in accordance with nearby sites on bedrock, indicating thedominance of surface loads in driving vertical motions away from areasaffected by local irrigation and groundwater fluctuations.

LETTER RESEARCH

0 0 M O N T H 2 0 1 4 | V O L 0 0 0 | N A T U R E | 3

Macmillan Publishers Limited. All rights reserved©2014

7. Fay, N. P., Bennett, R. A. & Hreinsdottir, S. Contemporary vertical velocity of thecentral Basin and Range and uplift of the southern Sierra Nevada. Geophys. Res.Lett. 35, L20309 (2008).

8. Bennett, R. A., Fay, N. P., Hreinsdottir, S., Chase, C. & Zandt, G. Increasing long-wavelength relief across the southeastern flank of the Sierra Nevada, California.Earth Planet. Sci. Lett. 287, 255–264 (2009).

9. Hammond, W. C., Blewitt, G., Li, Z., Plag, H. P. & Kreemer, C. Contemporary uplift ofthe Sierra Nevada, western United States, from GPS and InSAR measurements.Geology 40, 667–670 (2012).

10. Saleeby, J., Saleeby, Z. & Le Pourhiet, L. Epeirogenic transients related to mantlelithosphere removal in the southern Sierra Nevada region, California: Part II.Implications of rock uplift and basin subsidence relations. Geosphere 9, 394–425(2013).

11. Holzer, T. L. Elastic expansion of the lithosphere caused by groundwater depletion.J. Geophys. Res. 84, 4689–4698 (1979).

12. van Dam,T. et al.Crustal displacementsdue tocontinental water loading.Geophys.Res. Lett. 28, 651–654 (2001).

13. Jiang, Y., Dixon, T.H.&Wdowinski, S. Accelerating uplift in the NorthAtlantic regionas an indicator of ice loss. Nature Geosci. 3, 404–407 (2010).

14. Nof,R.N.et al.Risingof the lowestplaceonEarth due toDeadSeawater-level drop:evidence from SAR interferometry and GPS. J. Geophys. Res. 117, B05412 (2012).

15. Fu, Y.& Freymueller, J. T. Seasonal and long-term vertical deformation in theNepalHimalaya constrained by GPS and GRACE measurements. J. Geophys. Res. 117,B03407 (2012).

16. Heki, K. Snow loadandseasonal variation of earthquakeoccurrence in Japan.EarthPlanet. Sci. Lett. 207, 159–164 (2003).

17. Luttrell, K., Sandwell, D., Smith-Konter, B., Bills, B. & Bock, Y. Modulation of theearthquake cycle at the southern San Andreas fault by lake loading. J. Geophys.Res. 112, B08411 (2007).

18. Bettinelli, P. et al. Seasonal variations of seismicity and geodetic strain in theHimalaya inducedbysurfacehydrology.EarthPlanet.Sci. Lett.266,332–344(2008).

19. Gonzalez, P. J., Tiampo, K. F., Palano, M., Cannavo, F. & Fernandez, J. The 2011Lorca earthquake slip distribution controlled by groundwater crustal unloading.Nature Geosci. 5, 821–825 (2012).

20. Ben-Zion, Y. & Allam, A. A. Seasonal thermoelastic strain and postseismic effects inParkfield borehole dilatometers. Earth Planet. Sci. Lett. 379, 120–126 (2013).

21. Christensen, M. N. Late Cenozoic crustal movements in the Sierra Nevada ofCalifornia. Geol. Soc. Am. Bull. 77, 163–182 (1966).

22. Small, E. E. & Anderson,R. S. Geomorphically driven lateCenozoic rock uplift in theSierra Nevada, California. Science 270, 277–281 (1995).

23. Le Pourhiet, L. & Saleeby, J. Lithospheric convective instability could induce creepalong part of the San Andreas fault. Geology 41, 999–1002 (2013).

24. Blewitt, G., Kreemer, C., Hammond, W. C. & Goldfarb, J. M. Terrestrial referenceframe NA12 for crustal deformation studies in North America. J. Geodyn. 72,11–24 (2013).

25. Altamimi, Z., Collilieux, X. & Metivier, L. ITRF2008: an improved solutionof the international terrestrial reference frame. J. Geod. 85, 457–473(2011).

26. Schmidt, D. A. & Burgmann, R. Time-dependent land uplift and subsidence in theSanta Clara valley, California, from a large interferometric synthetic aperture radardata set. J. Geophys. Res. 108, 2416 (2003).

27. Argus, D., Fu, Y. & Landerer, F. Seasonal variation in total water storage in Californiainferred from GPS observations of vertical land motion. Geophys. Res. Lett. 41,1971–1980 (2014).

28. Dettinger, M. D. & Cayan, D. R. Large-scale atmospheric forcing of recenttrends toward early snowmelt runoff in California. J. Clim. 8, 606–623(1995).

29. California Department of Water Resources. California water plan update 2005.Volume 1: Strategic Plan. Dep. Water Res. Bull. 160–05 (2005).

30. Jeager, J., Cook, N. & Zimmerman, R. Fundamentals of Rock Mechanics 4th edn(Blackwell, 2007).

Acknowledgements Funding for this work comes from NSF EarthScope awardnumber EAR-1252210 to G.B. and W.C.H. GPS data were collected using theEarthScope Plate Boundary Observatory, SCIGN, BARGEN, BARD, CORS and IGSnetworks.We are particularly grateful toUNAVCO for operating thevastmajority ofGPSstations used in this project. GPS data were processed using the GIPSY OASIS IIsoftware and data products from the Jet Propulsion Laboratory.

Author Contributions C.B.A. and P.A. performed the analysis and wrote the paper.W.C.H. and G.B. analysed and processed the GPS data. All authors contributed to theinterpretations and preparation of the final manuscript.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to C.B.A. (colin.amos@wwu.edu).

RESEARCH LETTER

4 | N A T U R E | V O L 0 0 0 | 0 0 M O N T H 2 0 1 4

Macmillan Publishers Limited. All rights reserved©2014

METHODSGPS data processing. We use data from continuously recording GPS stations bet-ween 2122u and 2114u longitude and 34u to 38u latitude that had time series longerthan 2.5 years that were .50% complete (Extended Data Fig. 1). This data set in-cludes 566 continuous GPS stations within this geographic region. No episodiccampaign or semi-continuous stations were included. Percentage completeness isdefined as the number of daily solutions for the station divided by the number ofpotential daily solutions given the duration of the time series. Larger values for per-centage completeness are preferred in order to best resolve seasonal oscillations ofposition.

The data were processed as a part of a .11,000 station mega-network analysissystem that retrieves data daily and updates solutions weekly. We use the GIPSY/OASIS software provided by the Jet Propulsion Laboratory to estimate station coor-dinates every 24 h using the Precise Point Positioning method31. Ionosphere-free com-binations of carrier phase and pseudorange were obtained every 5 min. Calibrationswere applied for all antennas, ground receivers and satellite transmitters. To modeltropospheric refractivity, the Global Mapping Function was applied32, with tropo-spheric wet zenith delay and horizontal gradients estimated as stochastic random-walk parameters every 5 min (ref. 33). The observable model includes ocean tidalloading (including companion tides) coefficients supplied by Chalmers University34.Ambiguity resolution was applied to double differences of the estimated one-waybias parameters35, using the wide lane and phase bias method, which phase-connectsindividual stations to IGS stations in common view36. Satellite orbit and clock pa-rameters were provided by the Jet Propulsion Laboratory, who determine theseparameters in a global fiducial-free analysis using a subset of the available IGS corestations as tracking sites. Output station coordinates are initially in the loose frameof the Jet Propulsion Laboratory’s fiducial-free GPS orbits. These were transformedinto reference frame IGS08 using daily seven-parameter transformations suppliedby the Jet Propulsion Laboratory. IGS08 is derived from ITRF2008 (ref. 37) andconsists of 232 globally distributed IGS stations38. Finally, the solutions are alignedwith our custom reference frame (NA12) that co-rotates with stable North America24.This alignment is applied to each daily solution and provides a spatial filter to sup-press errors correlated at the continental scale. The mean formal uncertainty indaily vertical coordinate is ,3.0 mm. A summary reference table for GPS solutionmethodology is given at http://geodesy.unr.edu/gps/ngl.acn.The resulting time ser-ies are used to estimate rates of motion with respect to stable North America.GPS time series analysis. An outlier exclusion criterion was applied to remove dailysolutions that were more than 20 mm from the expected position based on a pro-visional time series model, or having position uncertainty greater than 20 mm. Foreach of the three components (east, north, up) of each GPS time series we solved forthe intercept b, rate v, steps at the times teqp of N known equipment changes thatcan introduce discontinuities of size D. We solve for annual and semiannual sea-sonal terms by solving for the cosine (Ci) and sine (Si) terms. For example, the eastcomponent is:

e tð Þ~bzvtzXN

i~1

DiH t{teqp,i� �

zX2

k~1

½Ckcos 2pktÞzSksin 2pktð Þ�ð

whereH tð Þ is the Heaviside step function, where t is in years. We use a dampedlinear inversion with weights on the prior uncertainties of the model parametersscaled to allow the expected range of values for each parameter. However, becauseonly continuously recording GPS stations were used, all parameters in the estima-tion (including seasonal terms) were well constrained, so the damping parametershad little influence on the solutions. The amplitude of the annual terms werecalculated with:

A~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2zS2ð Þ

pand the day of year from:

day~3652p

� �tan{1 S

C

� �

Finally, stations were excluded if their rate uncertainty was greater than 1 mm yr21

or their rate was clearly anomalous (that is, greater than 100 mm yr21). In all, weestimated solutions for 566 stations, 96% of which have time series duration greaterthan 4.0 years, and 92% of which are 80% or more complete (Extended Data Fig. 2).Example vertical GPS time series are shown in Extended Data Fig. 3. GPS time seriesparameters, including vertical uplift rate and the amplitude and timing of peakuplift are included as source data for Fig. 1.Elastic half-space model. We use an elastic half-space model for calculating theelastic response of the crust to changes in total water storage. The model is two-dimensional and accounts for surface and subsurface deformation induced by a lineload over the surface of the half-space along a finite-width strip (half-width a,

Extended Data Fig. 4a). Model derivation can be found in ref. 30, with a smallcorrection from ref. 13. The normal displacement of the surface (z 5 0) along aprofile perpendicular to the load centred at x 5 xd is given by

u x,z~0ð Þ~ 1{nð ÞN0

pG 2að Þ

2az x{xd{að Þ ln x{xd{aj j{ x{xdzað Þ ln x{xdzaj j½ �zK

where x and z are the horizontal and vertical dimensions, respectively, N0 is a lineload, n is Poisson’s ratio, G is the shear modulus, a is the strip half-width and K is afar-field rate offset correction. We fix the position of the line load to the centre ofthe San Joaquin Valley, and use a load width of 2a 5 60 km corresponding to theaverage width of the valley floor. Given a range of elastic parameters (n5 0.25,G 5 35 6 5 GPa), we can fit the vertical velocities derived from the GPS data by aline-load rate with _N0 5 8.8 (61.3) 3 107 N m21 yr21 (see Fig. 2). Goodness of fitis estimated using a reduced chi-square criterion. Our best-fit model has a reducedchi-square of 1.9, indicating a reasonably good fit.Unloading from estimated mass loss. The combined Sacramento–San JoaquinRiver basins lost an estimated volume of 30.9 (62.6) km3 in total water storage overa 6.5-year period between October 2003 and March 2010 (ref. 3). Of this volume,20.3 (63.8) km3 is attributed to loss of groundwater in the Central Valley, 80% ofwhich occurred in the San Joaquin Valley. Assuming that changes in total waterstorage not related to groundwater (for example, soil moisture, surface water andsnow) are distributed equally between the Sacramento and San Joaquin basins, thetotal water storage change in the San Joaquin Valley over the 6.5-year period is21.5 (66.2) km3. This change is equivalent to a rate of mass loss of 3.3 (61.0) 3 103

gigatons per year, and therefore a rate of unloading of 3.2 (61.0) 3 1013 N yr21

distributed over the 450 km length of the valley. Taking into account the range ofuncertainties in groundwater and total water storage change3, the equivalent line-load rate is thus _N0 5 7.2 (62.1) 3 107 N m21 yr21, in agreement with the estimatefrom the elastic model based on the vertical GPS velocities. If we add the estimated1860–2003 depletion of 140 km3 (ref. 4) to the 2003–2010 water loss, the cumulativehistoric load decrease amounts to ,3.5 3 109 N m21.Stress modelling. The two-dimensional stress components at any point at depthcaused by a distributed line load N0 (in N m21, the negative value indicates unload-ing) at the surface of the elastic half-space are calculated as30:

txx~N0

2pah1{h2ð Þz sin h1{h2ð Þ cos h1zh2ð Þ½ �

tzz~N0

2pah1{h2ð Þ{ sin h1{h2ð Þ cos h1zh2ð Þ½ �

txz~N0

2pasin h1{h2ð Þ sin h1zh2ð Þ½ �

where h1 and h2 are the angles from both edges of the load measured clockwisefrom the positive x direction, a is the load half-width and z is positive downward(Extended Data Fig. 4a). The shear and normal stress components resolved on afault plane dipping at an angle Q with a strike direction normal to the xz plane aregiven by:

tn~tzz cos2 Qð Þ{2txz sin Qð Þ cos Qð Þztxx sin2 Qð Þ

tz~ tzz{txxð Þ sin Qð Þ cos Qð Þztxz cos2 Qð Þ{ sin2 Qð Þ� �

Using Coulomb failure assumptions30, the Coulomb failure stress is expressed as:

sc~ tzj jzm tn{pð ÞzS

where m is the coefficient of friction, p is pore-fluid pressure and S is cohesion.Assuming that p, S and m are constant over time, the change in Coulomb stress isgiven by39:

Dsc~D tzj jzmDtn

where Dtn is positive for tension30,39. Similarly, the Coulomb stressing rate is given

by:

_sc~ _tzj jzm_tn

The first term on the right-hand side is valid for an isotropic failure plane. Weconsider the change in shear stress in the slip direction (or rake) and vary the sign

LETTER RESEARCH

Macmillan Publishers Limited. All rights reserved©2014

of the change in shear stress accordingly. For the San Andreas Fault we consider adip angle of 90u and the unclamping stress (or stress rate) is given simply by Dtxx

(or _txx). For the Coalinga blind thrust we use dip angles ranging between 15u and30u (refs 40 and 41) and a negative change in shear stress. We also consider a 65unortheast-dipping fault with a positive slip direction, simulating stress conditionson the Nunez thrust fault. For each fault we calculate the normal and shear stresscomponents at depths of 5 km and 15 km, representative of the seismogenic depthin Central California.Rate of unclamping of the San Andreas Fault. The rate of unclamping on a ver-tical fault (change in fault normal stress) can be calculated as30:

_txx~_N0

2pah1{h2ð Þz sin h1{h2ð Þ cos h1zh2ð Þ½ �

where h1 and h2 are the angles from both edges of the loading strip to any point atdepth, measured clockwise from the positive x direction. Taking an estimate of theunloading rate due to loss in water storage of 8.8 (6 1.3) 3 107 N m21 yr21, as wellas average spatial parameters for the San Andreas Fault (load width of 2a 5 60 km;distance to centre of the distributed line load of around 70 km; seismogenic depthsof 5–15 km), we estimate a rate of unclamping of 0.07–0.18 kPa yr21 at the seis-mogenic depth along the San Andreas Fault due to the elastic rebound caused byunloading (Extended Data Fig. 4b and c).

For the Coalinga blind thrust, we calculate the normal (tn) and shear (ts) stressrates resolved on a range of fault planes and depths. The Coulomb failure stress rate(ignoring pore pressure), _sc~ _tzj jzm_tn (ref. 30), is then calculated with m 5 0.7.Seasonal vertical displacements and stress changes. The elastic model can be usedto estimate total vertical displacements caused by variations in total water storage.The model considered is more broadly distributed over the Coast Range, the SanJoaquin Valley and the western Sierra Nevada (width of 200 km; Fig. 2), corres-ponding to the approximate total width of the San Joaquin River drainage basin.Taking an average of 200 mm of change in total water storage, we estimate a loadof ,4 3 108 N m21, which produces annual vertical displacements of 3–8 mm within

the load extent (Fig. 2). The associated stress variations on the San Andreas Fault areabout 1 kPa and the Coulomb stress changes on the Coalinga thrust are 1.0–1.7 kPa(Extended Data Fig. 4d and e). Varying the width of the distributed line loads by620 km and the load centre by 610 km has only a small impact (,0.5 kPa) on theresolved seasonal stress changes on the faults.

31. Zumberge, J. F., Heflin, M. B., Jefferson, D. C., Watkins, M. M. & Webb, F. H. Precisepoint positioning for the efficient and robust analysis of GPS data from largenetworks. J. Geophys. Res. 102, 5005–5017 (1997).

32. Boehm, J., Niell, A., Tregoning, P. & Schuh, H. Global mapping function (GMF): anew empirical mapping function based on numerical weather model data.Geophys. Res. Lett. 33, L07304 (2006).

33. Bar-Sever, Y. E., Kroger, P. M. & Borjesson, J. A. Estimating horizontal gradients oftropospheric path delay with a single GPS receiver. J. Geophys. Res. 103,5019–5035 (1998).

34. Scherneck, H.-G. A parametrized solid earth tide model and ocean tide loadingeffects for global geodetic baseline measurements. Geophys. J. Int. 106, 677–694(1991).

35. Blewitt, G. Carrier phase ambiguity resolution for the Global Positioning Systemapplied to geodetic baselines up to 2000 km. J. Geophys. Res. 94, 10187–10283(1989).

36. Bertiger,W.et al.Single receiverphaseambiguity resolutionwithGPSdata. J.Geod.84, 327–337 (2010).

37. Altamimi, Z., Collilieux, X. & Metivier, L. ITRF2008: an improved solution of theinternational terrestrial reference frame. J. Geodesy 85, 457–473 (2011).

38. Rebischung, P. et al. IGS08: The IGS realization of ITRF2008. GPS Solut. 16,483–494 (2012).

39. Harris, R. A. Stress triggers, stress shadows, and implications for seismic hazard.J. Geophys. Res. 103, 24347–24358 (1998).

40. Stein, R. S. & Ekstrom, G. E. Seismicity and geometry of a 110-km long blind thrustfault. 2. Synthesis of the 1982-1985 California earthquake sequence. J. Geophys.Res. 97, 4865–4883 (1992).

41. Toda, S. & Stein, R. S. Response of the San Andreas fault to the 1983 Coalinga-Nunez earthquakes: an application of interaction-based probabilities. J. Geophys.Res. 107, 2126 (2002).

RESEARCH LETTER

Macmillan Publishers Limited. All rights reserved©2014

Extended Data Figure 1 | Vertical GPS data. Map of vertical uplift rates spanning California and the western Great Basin, including stations within the CentralValley groundwater basin. El., elevation; GPS vert., GPS vertical velocity.

LETTER RESEARCH

Macmillan Publishers Limited. All rights reserved©2014

Extended Data Figure 2 | GPS time series. The top panel shows a histogram ofthe duration of GPS time series. 96% are longer than 4.0 years. The bottom panelshows a histogram of the percentage of complete time series. 92% are greaterthan 80% complete.

RESEARCH LETTER

Macmillan Publishers Limited. All rights reserved©2014

Extended Data Figure 3 | GPS time series. Example vertical GPS timeseries for stations P056 and P570. Blue dots are daily vertical position solutions,red lines are the model estimated to represent the time series. The station

P056 is in the southern Central Valley near Porterville, California. The stationP570 is near Weldon, California, east of Lake Isabella.

LETTER RESEARCH

Macmillan Publishers Limited. All rights reserved©2014

Extended Data Figure 4 | Stress profiles from distributed line loads of anelastic half-space. a, Model setup, where _N0 is the line-load rate, a is the loadhalf-width, and h1 and h2 measure the angle downward from the positive xaxis to any point (x0, z0) at depth. _t and D _sc give the stress rates and Coulombstress rates, respectively. b, c, Stress rates at depths of 5 km (b) and 15 km (c)calculated from the long-term rate of unloading over the San Joaquin Valleywith a 5 30 km. Vertical dashed lines labelled SAF and CF represent thelocations of the San Andreas Fault and the Coalinga blind thrust faults(including the Nunez fault), respectively, relative to the load centre. Blackand coloured lines indicate stress components and Coulomb stress changes,respectively. The blue curves labelled N show Coulomb stress calculations for

favourably oriented faults with a 65u dip, representing the Nunez fault. Forthe red curve representing the Coalinga fault (CF), the calculations areperformed using a dip of 30u for unfavourably oriented faults. The greencurve represents unclamping stress for vertically dipping faults such as the SanAndreas Fault. d, e, Stress changes at depths of 5 km and 15 km from seasonal(peak-to-peak) load changes over the San Joaquin River Basin with a 5 100 km(full width of 200 km). The position of the San Andreas Fault and Coalingafaults reflects displacement of the load centre by 30 km relative to the long-termload. Varying the load half-width and the load centre by 610 km has onlya small impact (,0.5 kPa) on the resolved seasonal stress changes on thefaults.

RESEARCH LETTER

Macmillan Publishers Limited. All rights reserved©2014